Method of 3d printing a cellular solid

ABSTRACT

A method of printing a cellular solid by direct bubble writing comprises introducing an ink formulation comprising a polymerizable monomer and a gas into a nozzle, which includes a core flow channel radially surrounded by an outer flow channel. The ink formulation is directed into the outer flow channel and the gas is directed into the core flow channel. The ink formulation and the gas are ejected out of the nozzle as a stream of bubbles, where each bubble includes a core comprising the gas and a liquid shell overlying the core that comprises the ink formulation. After ejection, the polymerizable monomer is polymerized to form a solid polymeric shell from the liquid shell, and the bubbles are deposited on a substrate moving relative to the nozzle. Thus, a polymeric cellular solid having a predetermined geometry is printed.

RELATED APPLICATION

The present patent document is a continuation of U.S. patent applicationSer. No. 17/265,396, which was filed on Feb. 2, 2021, and is the U.S.national stage of PCT/US2019/044792, which was filed on Aug. 2, 2019,and which claims the benefit of priority under 35 U.S.C. § 119(e) toU.S. Provisional Patent Application No. 62/714,892, which was filed onAug. 6, 2018. All of the aforementioned patent applications are herebyincorporated by reference in their entirety.

TECHNICAL FIELD

The present disclosure is related generally to three-dimensionalprinting (3D printing) and more particularly to 3D printing of porousstructures or cellular solids.

BACKGROUND

With the growing need for lightweight, high-performance structuralmaterials, cellular solids have become increasingly relevant over thepast several decades. The microarchitecture of cellular solids provideshighly tunable functional properties, and thus they are ubiquitous innature and industry. Cellular solids found in nature may exhibit densitygradients, locally controlled cell sizes, and interconnectivities withincomplex three-dimensional (3D) shapes, which may allow functionalperformance to be optimized with minimal material use. For example, thestiff surface and porous core of bone and feathers provide a remarkableresistance to bending and crack propagation without sacrificing lowdensity and perfusability by blood vessels. Synthetic cellular solidshave numerous current and potential applications, such as thermalinsulation, battery electrodes, separation, scaffolds for artificialtissues, pressure sensors, and personal protective gear, owing to theirtunable mechanical properties, low density, and high surface-to-volumeratio.

The mechanical, thermal, acoustic, and electrical properties of cellularsolids are primarily defined by the porosity, the constituent material,and the interconnectivity between cells (i.e., open- vs. closed-cellarchitectures). Since observations from nature suggest that localcontrol of the micro-architecture may be a prerequisite for functionaloptimization, bulk techniques have been modified to fabricate gradedporous solids. However, these techniques are limited to molded partswith relatively uncontrolled cellular architectures. Additivemanufacturing has also been explored for fabricating cellular solids;however, current technologies may demonstrate an exclusivity betweenstructural control and build speed.

BRIEF SUMMARY

A method of printing a cellular solid with a predetermined micro- andmacro-architecture is described.

The method comprises, according to one embodiment, introducing an inkformulation and a gas into a nozzle, which includes a core flow channelradially surrounded by an outer flow channel. The ink formulation isdirected into the outer flow channel and the gas is directed into thecore flow channel. The ink formulation and the gas are ejected out ofthe nozzle as a stream of bubbles, where each bubble includes a corecomprising the gas and a liquid shell overlying the core that comprisesthe ink formulation. After ejection, the liquid shell is solidified toform a solid shell, and the bubbles are deposited on a substrate movingrelative to the nozzle. Thus, a cellular solid having a predeterminedgeometry is printed.

The method comprises, according to another embodiment, introducing anink formulation comprising a flowable polymer precursor, such as apolymerizable monomer, and a gas into a nozzle, which includes a coreflow channel radially surrounded by an outer flow channel. The inkformulation is directed into the outer flow channel and the gas isdirected into the core flow channel. The ink formulation and the gas areejected out of the nozzle as a stream of bubbles, where each bubbleincludes a core comprising the gas and a liquid shell overlying the corethat comprises the ink formulation. After ejection, the flowable polymerprecursor is cured to form a solid polymeric shell, and the bubbles aredeposited on a substrate moving relative to the nozzle. Thus, apolymeric cellular solid having a predetermined geometry is printed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic showing an exemplary bubble writing process wheregas-filled bubbles are ejected from a core-shell nozzle and deposited ona substrate at a spatially controlled location. After ejection, thebubbles are solidified such that a cellular solid or foam having apredetermined geometry is formed on the substrate.

FIGS. 2A-2D show images of bubble morphology as a function of gaspressure applied to the nozzle for a given ink flow rate, where acontinuous stream is ejected in FIG. 2A at low gas pressures, and atrain of monodisperse bubbles (FIG. 2B), a train of bi- or tridispersebubbles (FIG. 2C), and a spray of droplets (FIG. 2C) are ejected atincreasingly higher gas pressures.

FIG. 3A shows a phase diagram of the ejection regimes shown in FIGS.2A-2D as a function of the ink flow rate and the gas pressure.

FIG. 3B shows cell size distributions for cellular solids formed fromstreams of monodisperse and bidisperse bubbles.

FIG. 3C show cell diameter as a function of gas pressure for cellularsolids formed from streams of monodisperse and bidisperse bubbles.

FIGS. 4A-4C show an exemplary cellular solid printed from a stream ofmonodisperse bubbles at different magnifications to demonstrate thatboth the macroscopic architecture of the cellular solid, as well as themicro-architecture (e.g., cell size and uniformity), may be tightlycontrolled.

FIGS. 5A and 5B show formation mechanisms for cellular solids havingopen-cell (FIG. 5A) and closed-cell (FIG. 5B) architectures.

FIG. 6 is a plot of density of the cellular solid versus gas pressure,where the data symbols are identified in the legend of FIG. 3A.

FIG. 7A shows images of exemplary filaments formed by translation of thenozzle at a velocity of 35 mm/s, 70 mm/s, 150 mm/s, and 250 mm/s,respectively (left to right), at a gas pressure of 2.4 kPa. Eachfilament is made up of a plurality of bubbles, and the scale barsrepresent 1 mm.

FIG. 7B shows a plot of filament width versus translation velocity ofthe nozzle as a function of gas pressure.

FIGS. 8A-8D shows images of a cellular solid comprised of filamentsarranged in a logpile geometry that underwent printing at a pressure of4 kPa and a velocity of 60 mm/s. Scale bars represent 50 mm, 5 mm and0.5 mm, respectively.

FIGS. 9A and 9B show photographs of cellular solids of differentstiffness undergoing compression, and FIG. 9C shows a plot of Young'smodulus versus relative density for cellular solids having closed-celland open-cell architectures.

FIG. 10A is a schematic of conformal printing onto a mold to form a capcomprising stiff and soft regions.

FIG. 10B is a plot of force versus indentation depth for the cap of FIG.10A.

FIGS. 10C shows images and schematics of the cap acquiring differentshapes upon continued compression from the top, of which examples (i),(iii), and (iv) are stable, as reflected by the force-indentation curveshown in FIG. 10B.

FIG. 11 shows controlled compression of a cellular solid between twoelectrodes to measure electrical resistance as a function of compressivestress.

FIG. 12A shows resistance as a function of compressive stress forcellular solids having different elastic moduli, as indicated.

FIG. 12B shows sensitivity as a function of stress for the cellularsolids; the data reveal that sensitivity is inversely proportional tostress and collapses to a universal trend for all evaluated stiffnessvalues.

DETAILED DESCRIPTION

Described herein is a high-throughput additive manufacturing method inwhich a train of bubbles—each made up of a flowable materialencapsulating a gas—may be ejected from a nozzle towards a substrate andsolidified in-situ, allowing a foam or cellular solid to be fabricatedin a layer-by-layer 3D printing process that may be referred to asdirect bubble writing. The cellular solid may have a relative densityand an open- and/or closed-cell architecture that can be independentlyand locally controlled. As discussed below, the transition between openand closed cells within the cellular solid may be controlled by the gastype, and the relative density may be controlled by the gas pressure.The macroscopic shape of the cellular solid is also programmable due tothe nature of the bubble writing process; for example, bulk materials,filaments, lattices, shells, and out-of-plane pillars have beendemonstrated. Arbitrary shapes may also be formed. Control over themicro- and macroscopic architectures can provide tunability of the localmechanical, electrical, and transport properties of the cellular solids,enabling tailored materials for pressure sensing, sound control, heatexchange, catalysis, mechanical energy absorption and/or otherapplications.

Referring to FIG. 1 , the 3D printing or bubble writing method entails,according to one embodiment, introducing an ink formulation 102 and agas 104 into a nozzle 106 comprising a core flow channel 108 radiallysurrounded by an outer flow channel 110; the ink formulation 102 isdirected into the outer flow channel 110 and the gas 104 is directedinto the core flow channel 108. The ink formulation 102 and the gas 104are ejected out of the nozzle 106 as a stream of bubbles 112, where eachbubble 112 includes a core 114 comprising the gas 108 and a liquid shell116 overlying the core 114 that comprises the ink formulation 102. Afterejection, the liquid shell 116 is solidified to form a solid shell 118,such that the bubbles 112 are solidified, and the bubbles 112 aredeposited on a substrate 122 moving relative to the nozzle 106. One ormore layers of the bubbles 112 may be deposited to fabricate a cellularsolid 120 having a predetermined geometry.

The method may be applicable to any of a number of materials. The solidshells 118 and consequently the cellular solids 120 formed by directbubble writing may comprise a metal, ceramic, semiconductor and/orpolymer. The solid shells 118 of the bubbles 112 deposited on thesubstrate may serve as struts or walls (e.g., “cell walls”) of theresulting 3D printed cellular solids 120, which are formed from singleor multiple layers of contacting bubbles. The liquid shell 116 and theink formulation 102 may include a flowable precursor to the metal,ceramic, semiconductor or polymer formed as a consequence ofsolidification. For example, in the case of a metal, the flowableprecursor may comprise a molten metal or a dissolved metal salt. In thecase of a ceramic, the flowable precursor may comprise a preceramicpolymer or a suspension of ceramic particles. A suitable flowableprecursor for a semiconductor may comprise a metal alkoxide or anorganometallic precursor. In the example of a polymer, as discussed ingreater detail below, the flowable precursor, which may also be referredto as a prepolymer, may comprise a polymerizable monomer. The rheologyof the ink formulation may be controlled to permit an ink flow rate ofat least about 0.03 ml/min and as high as about 300 ml/min, such as fromabout 0.03 ml/min to about 100 ml/min, from about 0.1 ml/min to about 50ml/min, or, more typically, from about 3 ml/min to about 15 ml/min. Inaddition to the flowable precursor, the ink formulation and the liquidshell may include any additives deemed necessary or advantageous, suchas a surfactant, a solvent, a chemical curing agent, a crosslinkingagent, a photoinitiator, a thermal initiator, nanoparticles and/or ananoparticle precursor.

The solidifying may entail freezing, evaporating (e.g., jamming),curing, crosslinking and/or polymerizing. For example, to effectsolidification, the bubbles 112 may be exposed to a change intemperature (heating or cooling), light of a suitable wavelength, a dryatmosphere (to effect evaporation), or a chemical curing agent, such asa latent curing agent that may be included in the ink formulation. Inanother example, the liquid shell 116 may be solidified by exposure to acrosslinking agent in a liquid jet directed to impinge upon the ejectedbubbles 112. Solidification of the liquid shell 116 may occur prior to,during, or after deposition of the bubbles 112 on the substrate 122. Inother words, the bubbles 112 may be solidified prior to being depositedon the substrate 122; alternatively, the bubbles 112 may be solidifiedduring deposition, or only after being deposited on the substrate 122.

In a preferred embodiment, the method is employed to fabricate apolymeric cellular solid 120. Accordingly, referring again to FIG. 1 ,the method may include introducing an ink formulation 102 comprising aflowable polymer precursor and a gas 104 into a nozzle 106 comprising acore flow channel 108 radially surrounded by an outer flow channel 110,where the ink formulation 102 is directed into the outer flow channel110 and the gas 104 is directed into the core flow channel 108. The inkformulation 102 and the gas 104 are ejected out of the nozzle 106 as astream of bubbles 112, where each bubble 112 includes a core 114comprising the gas 104 and a liquid shell 116 overlying the core 114that comprises the polymerizable monomer. After ejection, the flowablepolymer precursor is cured (e.g., polymerized or crosslinked) to form apolymeric (solid) shell 118 from the liquid shell 116, such that thebubbles 112 are solidified. The bubbles 112 are deposited on a substratemoving relative to the nozzle and thus a polymeric cellular solid 120 isprinted. Local properties of the polymeric cellular solid 120 may bedetermined by the gas species and/or the gas pressure during printing,as explained below.

Curing may entail exposing the bubbles 112 to light, heat, or a chemicalcuring agent, for example. In a preferred embodiment, the flowablepolymer precursor comprises a polymerizable monomer (e.g., aphotopolymerizable monomer), and the curing entails exposing the bubbles112 to light, such as UV light, to effect polymerization. Suitableflowable polymer precursors may include polyepoxides, includingaliphatic epoxides, alicyclic polyepoxides, and aromatic polyepoxides.Monofunctional and/or polyfunctional meth(acrylate) or acrylatecontaining monomers, oligomers, and polymers are particularly useful.Bireactive polymerizable monomers, oligomers, or polymers, for example,a compound having at least one free-radically polymerizable group, andat least one epoxy group may also be useful. Either free radicalphotoinitiators or cationic photoiniatiators or photobases, orcombinations thereof may be used to initiate the polymerization of thepolymer precursor. Additional initiators, for example thermalinitiators, may also be included to further the extent of curing in anoven.

As indicated above, curing and thus solidification of the bubbles 112may occur before, during, or after deposition of the bubbles 112 on thesubstrate 122. In one example, curing (e.g., polymerization) of anejected bubble 112 may occur immediately after deposition on thesubstrate 122, such as within one second (1 s), within 0.5 s, within 0.2s, or within 0.05 s of deposition.

It is possible to form nanocomposite cellular solids 120 by dispersingmetal nanoparticles into the solid shell 118 of the bubbles 112. Thismay be achieved by, for example, incorporating a nanoparticle precursorsuch as a metal salt into the ink formulation 102 prior to bubblewriting, and then reducing the metal salt during solidification of theliquid shell 116, thereby forming metal nanoparticles dispersed in thesolid shell 118. For example, silver nanoparticles may be generatedwithin a polymeric shell 118 by UV-induced reduction of silver nitratethat is dissolved in the ink formulation 102. Suchnanoparticle-reinforced cellular solids 120 may be electrically and/orthermally conductive, and/or exhibit other properties imparted by thepresence of the nanoparticles. In one example discussed below, ananocomposite cellular solid 120 comprising metal nanoparticles isdeveloped and utilized as a pressure sensor.

The gas 104 employed for bubble writing may comprise mixtures of gasessuch as air, or other gases such as oxygen, nitrogen, helium, and/orargon. Typically, the gas 104 is directed into the nozzle 106 at apressure in a range from about 1 kPa to about 10 kPa, although a muchlarger range of gas pressures may be used (e.g., from about 0.1 kPa to1000 kPa) depending on the size of the nozzle and ink parameters.Printing experiments have revealed that bubble ejection from the nozzle106 may occur in four pressure-dependent regimes, as illustrated inFIGS. 2A-2D. At low gas pressures at a given ink flow rate (e.g., 10mL/min), a pure-liquid jet or stream is ejected from the nozzle, asshown in FIG. 2A. Increasing the pressure at the same flow rate resultsin a train of monodisperse bubbles, as shown in FIG. 2B, followed bybifurcation to bi- or tridisperse bubbles at higher pressures, as shownin FIG. 2C. At further increased pressures at the same flow rate, aspray of droplets and bubbles with poorly defined directionality may beobserved, as shown in FIG. 2D.

FIG. 3A is a plot of gas pressure P as a function of ink flow rate Qthat shows how the bubble morphology changes as a function of both ofthese parameters. The gas pressure may be understood to be the pressureat which the gas 104 is directed into the core flow channel 108. Theplot identifies suitable conditions to obtain a train of monodispersebubbles, which may be desirable for bubble writing. For example, thedata of FIG. 3A show that the following conditions may be suitable: agas pressure in a range from about 1.8 to 2.1 kPa at an ink flow rate ofabout 8 ml/min; a gas pressure in a range from about 2 to 3.2 kPa at anink flow rate of about 10 ml/min; a gas pressure in a range from about2.5 to 4 kPa at an ink flow rate of about 13 ml/min; and/or a gaspressure in a range from about 3.9 to about 4.5 kPa at a flow rate ofabout 15 ml/min. These values may depend on a number of factors, suchas: the liquid density, surface tension, and/or viscosity; the design ofthe nozzle; and/or the inner and outer diameter of the nozzle.Furthermore, additional bubble ejection regimes may be revealed fordifferent control parameters. For example, a continuous jet filled withmonodisperse bubbles is observed when using liquids with higherviscosities between 15 and 500 mPa s. Generally speaking, a train ofmonodisperse bubbles may be attained at a suitable ink flow rate for gaspressures P ranging from about 1.8 kPa to about 5 kPa. Typically, theink flow rate Q is in a range from about 7.5 ml/min to about 15 ml/min.The plot of FIG. 3A reveals that controlled bubble ejection may beobtained for the broadest range of gas pressures P at a flow rate Q=10mL/min; thus, this flow rate is employed for a number of experimentsdescribed in this disclosure.

A stream of monodisperse bubbles obtain at a suitable gas pressure andink flow rate may be deposited to form cellular solids with a uniformcell size and quasi-crystalline packing, as can be seen by the images ofFIG. 2B, where the top image shows monodisperse bubbles formed uponejection from the nozzle and the bottom image shows the resultingcellular solid. The cell size distribution for the cellular solid ofFIG. 2B is plotted in FIG. 3B, revealing a distinct dominant celldiameter (with a coefficient of variation of 4%) that may be tuned from0.4 to 0.7 mm by increasing the gas pressure from 2.1 to 2.6 kPa, asshown in FIG. 3C. In general, the cells of the cellular solid may have anominal size in a range from about 0.01 mm to about 10 mm, depending onthe bubble writing parameters, with the range of about 0.1 mm to about 1mm, or from about 0.3 mm to about 0.7 mm, being typical. A benefit ofthe method is that the cells may have a locally adjustable size and alocally adjustable interconnectivity. For example, cellular solidshaving a gradient in the cell size (e.g., cells ranging in size fromsmall to large across all or a portion of the cellular solid) may befabricated.

In another example, bidisperse cellular solids including both large andsmall cell sizes are possible. Bidisperse cellular solids (FIG. 2C,bottom image) produced from a stream of bidisperse bubbles (FIG. 2C, topimage) are observed for gas pressures in the range 2.8≤P≤3.4 kPa. Thebidisperse cellular solid of FIG. 2C has two dominant cell diameters of0.3±0.1 and 0.7±0.1 mm, as can be observed in the top image of FIG. 2Calong with the data of FIGS. 3B and 3C. At higher pressures, atri-disperse distribution is observed for P=3.6 kPa, followed by a broadcell size distribution at P=4.8 kPa.

An exemplary cellular solid that is printed from a stream ofmonodisperse bubbles is shown at different magnifications in FIGS. 4A to4C to demonstrate that both the macroscopic architecture of the cellularsolid, as well as the micro-architecture (e.g., cell size anduniformity), may be tightly controlled. The macroscopic architecture ofthe cellular solid may be influenced by the path taken by the nozzlerelative to the substrate as bubbles are ejected, as discussed furtherbelow. The length scale of both the individual pores or cells and themacroscopic shape are generally controlled by the bubble size anddeposition location, respectively, but it may be further tuned by using“shrinking” polymers. For example, an ink formulation that includes asubstantial amount of solvent (water) has been observed to decrease byabout 30% in size upon evaporation of the solvent from the polymershell.

The cellular solid 120 may have an open-cell or closed-cellarchitecture. Open cell architectures may be formed by using air as thegas to fill the bubbles. This is because oxygen inhibits polymerizationof the liquid shells 116 of the bubbles 112 which in turn become thecell walls 124 of the cellular solid 120, as illustrated in FIG. 5A. Incontrast, closed-cell architectures may be formed if nitrogen is used asthe gas, which leads to polymerization of intact cell walls 124, asillustrated in FIG. 5B.

Specifically, oxygen can penetrate the surface layer of the still-liquidcell walls and inhibit polymerization of an acrylate-based ink, as shownin the examples. The oxygen penetration depth is estimated asδ=(Dt_(s))^(1/2)≈20 μm, where D≈2·10⁻⁹ m²s⁻¹ denotes the oxygendiffusion coefficient, and is t_(s)≈200 ms refers to the solidificationtime scale estimated using high-speed imaging. As the struts or wallsbetween closed cells are typically thinner than 40 μm, oxygen caninhibit polymerization over the entire thickness, causing the liquidwalls to eventually rupture. The struts have a typical thicknessexceeding 100 μm, so only their surface may be oxygen-inhibited; thecore may polymerize into a solid skeleton which constitutes an open-cellcellular solid as illustrated in FIG. 5A. In contrast, when oxygen isreplaced with nitrogen, the struts or walls between cells entirelypolymerize upon UV-exposure, resulting in closed-cell cellular solids asshown in FIG. 5B. Interestingly, the open-cell cellular solid shown inthe microscope image of FIG. 5A has a similar density (131±5 kg/m³) tothat of the closed-cell cellular solid (122±5 kg/m³) shown in themicroscope image of FIG. 5B. Generally, the transition between open-celland closed-cell architectures may be determined by competition betweenthe rupture of the walls of the liquid bubbles (pores or cells) and thesolidification time. Alternative methods to control the solidificationrate (and thus solidification time) include the UV light intensity, thetype and amount of photo-initiator, the type and amount of monomers orpolymers, the temperature, the type and amount of chemicals that inducesolidification, and the wall thickness. The rupture time can be tunedpassively, for example, by changing the liquid viscosity or surfacetension, the wall thickness (i.e., the gas-to-liquid ratio), theparticle size in suspensions, the droplet size in emulsions, and/or thepressure of the surrounding environment. The rupture time can also bereduced actively, for example by rupturing the walls with a focusedlaser, vibrations, or mechanical puncture. Combined, these parameterscan be tuned to control the transition between open- and closed cellfoams.

The gas pressure may also influence the relative density, which may beexpressed as % density (or, inversely, the fraction of porosity) of acellular solid formed by bubble writing. At very low pressures, thecellular solid may be 100% dense (100% solid; no cells), while at muchhigher pressures, the cellular solid may be 10% or less dense (less than10% solid fraction, where the gas-filled cells make up 90% or more ofthe cellular solid). Depending on the gas pressure, the cellular solidmay be as low as 1% dense, or as low as 0.1% dense. The relationshipbetween gas pressure and density is illustrated by the data of FIG. 6 ,which shows density of the cellular solid as a function of gas pressureapplied to the nozzle, where the data symbols are identified by thelegend of FIG. 3A. The data show, for example, that an all-solidcellular solid having a density ρ_(F)=1100 kg m⁻³ may be formed at a gaspressure P≤1.9 kPa, while a cellular solid having a 10% solid fractionρ_(F)=115 kg m⁻³ may be formed at a gas pressure P=4.4 kPa.

The cellular solid 120 may have any three-dimensional macroscopicarchitecture that can be formed by x-, y-, and/or z- motion of thenozzle 106 relative to the substrate 122 during bubble ejection. Aswould be recognized by the skilled artisan, to achieve relative motionbetween the substrate 122 and the nozzle 106, one or both of thesubstrate 122 and the nozzle 106 may be moved. In other words, thesubstrate 122 may remain stationary while the nozzle 106 is moved, thenozzle 106 may remain stationary while the substrate 122 is moved, orboth of the substrate 122 and the nozzle 106 may be moved. Typically, anozzle 106 suitable for bubble writing may have sub-millimetric (e.g.,less than about 1 mm) internal dimensions and thus may be classified asa millifluidic or microfluidic device. For example, the exemplarynozzles 106 employed for the experiments described in this disclosurehave internal dimensions (e.g., a nozzle opening or outlet) in the rangefrom about 200 microns to about 500 microns.

In one implementation, typical velocities of the nozzle 106 relative tothe substrate 122 range from about 1 mm/s to about 300 mm/s. For printvelocities V greater than 20 mm/s, filaments made up of bubbles may beformed as shown in FIG. 7A. The exemplary filaments shown from left toright in FIG. 7A are formed by translating the nozzle operated at a gaspressure of 2.4 kPa at a velocity of 35 mm/s, 70 mm/s, 150 mm/s, and 250mm/s, respectively. The data of FIG. 7B show that the width of thefilament increases for increasing pressures or decreasing x-y velocitiesof the nozzle.

Bubble writing may be carried out in air or in a controlled environment(e.g., oxygen, nitrogen, helium, and/or argon) at atmospheric pressureor at a reduced pressure (e.g., vacuum conditions).

As the bubbles are solidified (e.g., within 0.2 second after impact ordeposition), they may be readily stacked into large, multi-scalecellular solids, as shown for example in FIGS. 8A-8D. This exemplarycellular solid, with outer dimensions of 60×40×3 cm³, was printed inonly 22 minutes, as the liquid flow rate of 10 mL/min was increased toabout 80 mL/min by the entrainment of air. In another example, verticalfilaments or pillars of controlled height may be fabricated byimmobilizing the printhead (nozzle) for a fixed duration as bubbles areejected. Moving the printhead at low x-y velocity, e.g., V<14 mm/s, mayenable control over the inclination angle of these pillars with respectto the substrate. Bridges may be formed by touching pillar tipsout-of-plane. The transition between inclined pillars to horizontalfilaments is observed to occur at a velocity V=15±0.5 mm s⁻¹, at whichhorizontal overhangs (90°) may be formed.

Combining tunable micro-architectures, as shown for example in FIGS. 5Aand 5B, with programmable 3D macro-architectures, as shown for examplein FIG. 8A, enables cellular solids with locally tailorable mechanicalproperties to be fabricated. For example, stiffness may be controlled asillustrated in FIGS. 9A-9C, where the photographs show the behavior of asoft cellular solid and a stiff cellular solid, respectively, uponapplication of a 100 g mass. The elastic modulus E (stiffness) may betuned over multiple orders of magnitude, as shown by the data of FIG.9C, by varying the relative density of a cellular solid prepared bybubble writing. The soft and stiff cellular solids of FIGS. 9A and 9Bhave open-cell architectures at different relative densities (ρ_(rel)).The data in FIG. 9C cover the relative density range of 0.1<ρ_(rel)<1,but it is recognized that the relative density of the cellular solidsmay be controlled over a much larger range, e.g., 0.001<ρ_(rel)<1, viabubble writing.

Theory predicts a power-law

${\frac{E}{E_{0}} \sim \left( \rho_{rel} \right)^{n}},$

where

$\rho_{rel} = \frac{\rho}{\rho_{0}}$

is the relative density and ρ₀ and E₀ denote the bulk density andelasticity, respectively. An exponent n=2 is predicted and widelyobserved for open-cell solids, whereas 1<n<2 is predicted forclosed-cell solids with increasingly thin walls. Although these valuesare derived for ρ_(rel)<0.1, they are usually still accurate at higherdensities for a wide range of solids. For closed-cell foams, a value ofn≈2 is observed. This high value indicates that the faces aresignificantly contributing to the stiffness, which is hardly surprisingin view of their low thickness. For open-cell solids, n≈4 is observed.This high value may be attributable to relatively thin struts formed dueto partial oxygen inhibition during polymerization. Thus, for cellularsolids formed by direct bubble writing, values of n may range from about2≤n≤4. In future implementations, this exponent may decrease to lowervalues, but may not be lower than 1.

High values of n indicate exceptional stiffness tunability over amoderate density range, which is exploited for conformal printing of anexemplary tri-stable cap with stiff and soft regions, as illustrated inFIG. 10A. As shown in FIG. 10C, this cap snaps into different shapesupon continued compression from the top, of which examples (i), (iii),and (iv) are stable as reflected by the force-indentation curve shown inFIG. 10B. As such gradients in stiffness can be applied at any location,direct bubble writing enables one-step manufacture of 3D-architecturedmaterials that bend or cushion in a locally optimized fashion.

As indicated above, nanocomposite cellular solids 120 may be formed bydispersing nanoparticles (e.g., metal nanoparticles) in the cell walls124. In one example, a nanocomposite cellular solid including silvernanoparticles is developed and utilized as a pressure sensor. It isfound that the nanocomposite cellular solid exhibits an elastic moduluscomparable to a cellular solid prepared under the same conditions butwithout nanoparticle reinforcement. To determine the influence of themetal nanoparticles on the conductivity, electric resistance is measuredby controlled compression between two electrodes, as illustrated in FIG.11 , for nanocomposite cellular solids having different elastic moduli.The resistance data shown in FIG. 12A reveal monotonically decreasingvalues of resistance as a function of compressive stress and stiffness.To assess the sensing performance of these cellular solids, thesensitivity

$S = {{\left( \frac{dR}{R} \right)/d}P}$

is determined as shown in FIG. 12B. The inversely proportionaldependency on the compressive stress indicates that the pressure sensingerror is constant over the full material and pressure range. Directbubble writing enables one-step fabrication of pressure sensors withcontrolled shape and tunable stiffness for a stress range that includesboth gentle touch (<10 kPa) and object manipulation (10 to 100 kPa).These sensors maintain their elasticity under moderate compression, andthus can be used repeatedly. For example, low-density cellular solids(e.g., where

$\rho_{rel} = \frac{\rho}{\rho_{0}}$

is about 0.2 or less) may exhibit elastic behavior up to 60% strain inthe 0.2 to 20 kPa range, and high-density cellular solids (e.g., whereρ_(rel) is around 0.4, such as from 0.3 to 0.5) may be elastic up to 40%strain over a large stress range from 0.5 to 100 kPa.

Materials and Methods

Ink formulation: Poly(ethyleneglycol)diacrylate (M_(n) 700) (Sigma),Tween 80 (Sigma), Irgacure 651 (BASF), deionized water, and nitrogen areobtained at the highest purity available and used without furtherpurification unless otherwise specified. For 100 g of the inkformulation: PEG-DA (35 g), Tween 80 (2 g) and Irgacure 651 (0.4 g) arecombined and mixed using Flacktek speed mixer for 10 minutes at 2350rpm. Deionized water (62.6 g) is then stirred into the mixture for 30seconds. The resulting ink formulation is then kept from light toprevent photopolymerization prior to usage. A conductive ink formulationfor use in printing a nanocomposite cellular solid is prepared by addingan additional 10 g of a nanoparticle precursor, specifically, silvernitrate (a metal salt), to the mixture. Subsequently, the inkformulation (with or without the metal salt addition) is purged withnitrogen for 20 minutes prior to usage. The syringes are filled with theink formulation in an oxygen-free atmosphere.

Ink and gas supply: The ink formulation is supplied with a syringe pump(Harvard Apparatus), on which two 60-mL plastic syringes(Becton-Dickinson) containing the ink formulation are mounted. Theirflows (5 mL min⁻¹ per syringe) are combined by a T-junction and suppliedto the nozzle with standard PEEK tubing and Luer-lok components (IDEXHealth&Science). This tubing is sufficiently long and flexible to bridgethe gap between the syringe pump and the moving printhead (nozzle). Thegas pressure is controlled with a computer-controlled pressure box(Alicat PC-15PSIG-D). Either house air or nitrogen tanks (AirGas) areused.

Print process: A dedicated printhead is employed for direct bubblewriting. A nozzle and ends of a splitting optical fiber are mounted ontoan automated 3D-stage (Aerotech), of which the motion path is programmedin G-code (or RS-274). Disposable core-shell nozzles suitable for bubblewriting are 3D-printed using an Envisiontec Aureus printer. The nozzleinlets are connected to PEEK tubing (IDEX Health & Science) usingstandard Luer-lok components (IDEX Health&Science). Inside the nozzle,the core flow channel and the outer flow channel are concentricallyaligned. Nozzles with inner and outer diameters of 0.30±0.02 mm and0.44±0.02 mm, respectively, are selected after printing to minimizevariability. UV light is provided by an Omnicure light source (OmnicureS2000, Excelitas technologies), to which a splitting optical fiber isconnected. The four ends of this fiber are pointed towards the bubbleimpact location, providing a relatively homogeneous intensity of 0.8±0.2mW cm⁻² over a circular area with a diameter of 5 cm at the depositionplane (at a length L of about 10 cm from the nozzle).

Imaging: A live view of the train of bubbles during printing is providedby a Q-click F-M12 camera (Qimaging) set to a shutter time of 30 μs.High-speed videos are obtained by a V7.1 (Phantom) camera operated at6000 frames per second. Illumination for the camera is provided with astandard light source (Thorlabs, OSL 2).

Print path: The cellular solid shown in FIGS. 8A and 8B is printed in alog pile pattern. For typical samples, the distance between lines in thelog pile is from about 4 mm to 7 mm, and the printing velocity is set to40 mm s⁻¹ (for low-pressures) to 100 mm s⁻¹ (for high pressures). Thez-position of the nozzle is not adjusted during printing of thesesamples. Vertical pillars are formed by immobilizing the printhead.Inclined pillars are printed by moving the printhead at V<10 mm s⁻¹.V-shaped bridges are made by printing two inclined pillars from theirbase and connecting their tips. The spherical cap of FIGS. 10A-10C isprinted onto a mold, as shown in FIG. 10A, using an approach that may bereferred to as conformal printing. The cap is manually removed from themold after fabrication.

Post-curing: For nanocomposite cellular solids, the top and bottom sidesof printed samples are exposed to broadband UV light as provided by theOmnicure UV source, for 10 minutes per side, to enhance the formation ofnanoparticles.

Drying: After printing, the cellular solids are stored in a dryingcabinet to which water-absorbing grains (Drierite, VWR) are added andregularly replaced (typically once per two days) to keep the humiditybetween 70% and 85%. Humidity below 70% is observed to result incracking, especially for relatively dense samples and closed-cellcellular solids. After several days, the cellular solids are fully dry(up to 3 weeks for the larger closed-cell samples), as observed from asudden drop of the humidity in the chamber to 30% or less. The size andweight are measured before and after drying for 18 samples thatunderwent direct writing at gas pressures in the range from 2.2 kPa to5.6 kPa. Virtually isotropic shrinkage is observed, at 27.8±1.1% in x-ydirection and 28.0±2.1% in z-direction. Combined, shrinkage and massloss during drying (59±1%) result in a density increase of 9%.

Although the present invention has been described in considerable detailwith reference to certain embodiments thereof, other embodiments arepossible without departing from the present invention. The spirit andscope of the appended claims should not be limited, therefore, to thedescription of the preferred embodiments contained herein. Allembodiments that come within the meaning of the claims, either literallyor by equivalence, are intended to be embraced therein.

Furthermore, the advantages described above are not necessarily the onlyadvantages of the invention, and it is not necessarily expected that allof the described advantages will be achieved with every embodiment ofthe invention.

1. A method of printing a cellular solid, the method comprising:introducing an ink formulation and a gas into a nozzle comprising a coreflow channel radially surrounded by an outer flow channel, the inkformulation being directed into the outer flow channel and the gas beingdirected into the core flow channel; ejecting the ink formulation andthe gas out of the nozzle as a stream of bubbles, each bubble includinga core comprising the gas and a liquid shell overlying the core andcomprising the ink formulation; after ejection, solidifying the liquidshell to form a solid shell, the bubbles thereby being solidified withina timeframe from prior to deposition of the bubbles on the substrateuntil to within one second after deposition of the bubbles on thesubstrate; and depositing the bubbles on a substrate moving relative tothe nozzle according to a predetermined geometry, thereby printing asolid cellular structure having said predetermined geometry.
 2. Themethod of claim 1, wherein each of the solid shells comprises a metal,ceramic, semiconductor, and/or polymer.
 3. The method of claim 1,wherein solidifying comprises freezing, evaporating, curing,crosslinking, and/or polymerizing.
 4. The method of claim 1, wherein thestream of bubbles is a monodisperse stream of bubbles.
 5. The method ofclaim 1, wherein the ink formulation further comprises a nanoparticleprecursor.
 6. The method of claim 5, wherein the nanoparticle precursorcomprises a metal salt, and wherein, during solidification of the liquidshell, the metal salt is reduced to form metal nanoparticles dispersedin the solid shell.
 7. The method of claim 1, wherein the gas isselected from the group consisting of: air, oxygen, nitrogen, helium andargon.
 8. The method of claim 1, wherein the gas is directed into thenozzle at a pressure in a range from about 1 kPa to about 10 kPa.
 9. Themethod of claim 1, wherein a flow rate of the ink formulation is in arange from about 3 ml/min to about 15 ml/min.
 10. The method of claim 1,wherein the cellular solid comprises a closed cell microarchitecture.11. The method of claim 1, wherein the cellular solid comprises an opencell microarchitecture.
 12. The method of claim 1, wherein cells of thecellular solid have a nominal size in a range from about 0.01 mm toabout 10 mm.
 13. The method of claim 1, wherein the nozzle movesrelative to the substrate at a translation speed in a range from about 1mm/s to about 300 mm/s.
 14. The method of claim 1, wherein the cellularsolid is configured for pressure sensing, sound control, heat exchange,catalysis, and/or mechanical energy absorption.
 15. A method of printinga cellular solid, the method comprising: introducing an ink formulationcomprising a flowable polymer precursor and a gas into a nozzlecomprising a core flow channel radially surrounded by an outer flowchannel, the ink formulation being directed into the outer flow channeland the gas being directed into the core flow channel; ejecting the inkformulation and the gas out of the nozzle as a stream of bubbles, eachbubble including a core comprising the gas and a liquid shell overlyingthe core comprising the ink formulation; after ejection, curing theflowable polymer precursor within a timeframe from prior to depositionof the bubbles on the substrate until to within one second afterdeposition of the bubbles on the substrate to form a solid polymericshell from the liquid shell, the bubbles thereby being solidified; anddepositing the bubbles on a substrate moving relative to the nozzleaccording to a predetermined geometry, thereby printing a solid cellularstructure having said predetermined geometry.
 16. The method of claim15, wherein curing the flowable polymer precursor comprises exposing thebubbles to light, heat or a chemical curing agent.
 17. The method ofclaim 15, wherein the flowable polymer precursor comprises apolymerizable monomer.
 18. The method of claim 17, wherein thepolymerizable monomer is a photopolymerizable monomer, and wherein thepolymerizing comprises exposing the bubbles to ultraviolet (UV) light.19. The method of claim 15, wherein the ink formulation includes ananoparticle precursor.
 20. The method of claim 19, wherein thenanoparticle precursor comprises a metal salt, and wherein, duringcuring of the liquid shell, the metal salt is reduced to form metalnanoparticles dispersed in the solid shell.
 21. The method of claim 15,wherein the stream of bubbles is a monodisperse stream of bubbles. 22.The method of claim 15, wherein the gas is selected from the groupconsisting of: air, oxygen, nitrogen, helium and argon.
 23. The methodof claim 15, wherein the gas is directed into the nozzle at a pressurein a range from about 1 kPa to about 10 kPa.
 24. The method of claim 15,wherein a flow rate of the ink formulation is in a range from about 3ml/min to about 15 ml/min.
 25. The method of claim 15, wherein thecellular structure comprises a closed cell microarchitecture.
 26. Themethod of claim 15, wherein the cellular structure comprises an opencell microarchitecture.
 27. The method of claim 15, wherein pores of thecellular structure have a nominal size in a range from about 0.3 mm toabout 0.7 mm.
 28. The method of claim 15, wherein the nozzle is movedrelative to the substrate at a translation speed in a range from about 1mm/s to about 300 mm/s.
 29. The method of claim 15, wherein the cellularsolid is configured for pressure sensing, sound control, heat exchange,catalysis, and/or mechanical energy absorption.
 30. The method of claim15, wherein the cellular solid comprises locally controlled gradients inpore size, interconnectivity, and/or material composition.